Introduction to Differential Equations
This worksheet introduces fundamental concepts and methods for solving basic differential equations suitable for Grade 11 students.
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Standards
Introduction to Differential Equations
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Read each question carefully and provide your answers in the space provided. Show all your work for full credit.
A differential equation is an equation that relates one or more functions and their derivatives. They are used to model various phenomena in science, engineering, economics, and more.
1. Which of the following is a differential equation?
$$y = 3x^2 + 2x - 5$$
$$\frac{dy}{dx} = 4x$$
$$2x + 5 = 11$$
$$f(x) = \sin(x)$$
2. The order of a differential equation is determined by the:
Highest power of the dependent variable
Highest derivative in the equation
Number of terms in the equation
Degree of the equation
3. A differential equation that contains only ordinary derivatives with respect to a single independent variable is called an differential equation.
4. The process of finding a function that satisfies a given differential equation is called the differential equation.
5. Determine the order of the following differential equation: $$\frac{d^3y}{dx^3} + 5\frac{dy}{dx} - 2y = e^x$$
6. Is the following equation a linear or non-linear differential equation? Explain your reasoning. $$\frac{dy}{dx} + xy^2 = \sin(x)$$
Match each term on the left with its definition on the right.
7. General Solution
a. A solution that satisfies specific initial conditions.
8. Particular Solution
b. A solution containing arbitrary constants.
9. Initial Value Problem
c. A differential equation along with additional conditions.
10. Find the general solution for the differential equation: $$\frac{dy}{dx} = 3x^2 - 2x + 1$$
11. Solve the initial value problem: $$\frac{dy}{dx} = 2x + 3$$ with $$y(0) = 5$$
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